Quantized symplectic actions and $W$-algebras

Abstract:

With a nilpotent element in a semisimple Lie algebra $\mathfrak {g}$ one associates a finitely generated associative algebra $\mathcal {W}$ called a $W$-algebra of finite type. This algebra is obtained from the universal enveloping algebra $U(\mathfrak {g})$ by a certain Hamiltonian reduction. We observe that $\mathcal {W}$ is the invariant algebra for an action of a reductive group $G$ with Lie algebra $\mathfrak {g}$ on a quantized symplectic affine variety and use this observation to study $\mathcal {W}$. Our results include an alternative definition of $\mathcal {W}$, a relation between the sets of prime ideals of $\mathcal {W}$ and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of $\mathcal {W}$ in the case of classical $\mathfrak {g}$ and the separation of elements of $\mathcal {W}$ by finite-dimensional representations.